LMIs in Control/pages/Delay Dependent Time-Delay Stabilization

Stabilization of Time-Delay Systems - Delay Dependent Case

Suppose, for instance, there was a system where a time-delay was introduced. In that instance, stabilization would have to be done in a different manner. The following example demonstrates how one can stabilize such a system while being dependent on the delay.

The System

For this particular problem, suppose that we were given the time-delayed system in the form of:

${\begin{aligned}{\begin{cases}{\dot {x}}(t)&=Ax(t)+A_{d}(t-d)+Bu(t),\\x(t)&=\phi (t),t\in [0,d],0<d\leq {\bar {d}},\\\end{cases}}\end{aligned}}$

where

${\begin{aligned}&A{\text{, }}{A_{d}}\in \mathbb {R} ^{n\times n}{\text{, }}{B}\in \mathbb {R} ^{n\times r}{\text{ are the system coefficient matrices,}}\\&\phi (t){\text{ is the initial condition,}}\\&d{\text{ represents the time-delay, and}}\\&{\bar {d}}{\text{ is a known upper-bound of }}d\\\end{aligned}}$

Then the LMI for determining the Time-Delay System for the Delay-Dependent case would be obtained as described below.

The Data

In order to obtain the LMI, we need the following 3 matrices: $A{\text{, }}{A_{d}}{\text{, and }}B$.

The Optimization Problem

Suppose - for the time-delayed system given above - we were asked to design a memoryless feedback control law where $u(t)=Kx(t)$ such that the closed-loop system:

${\begin{aligned}{\begin{cases}{\dot {x}}(t)&=(A+BK)x(t)+A_{d}(t-d),\\x(t)&=\phi (t),t\in [0,d],0<d\leq {\bar {d}},\\\end{cases}}\end{aligned}}$

is simultaneously both uniform and asymptotically stable, then the system would be stabilized as follows.

The LMI: The Delay-Dependent Stabilization of Time-Delay Systems

From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a scalar $0<\beta <1$, a symmetric matrix $X>0$ and a matrix $W$ that satisfy the following:

${\begin{aligned}{\begin{bmatrix}\Phi (X,W)&&{\bar {d}}(X{A^{T}}+{W^{T}}{B^{T}})&&{\bar {d}}X{A_{d}^{T}}\\{\bar {d}}(AX+BW)&&-{\bar {d}}{\beta }I&&0\\{\bar {d}}{A_{d}}X&&0&&-{\bar {d}}(1-\beta )I\end{bmatrix}}&<0\\\end{aligned}}$

where$\Phi (X,W)=X{(A+{A_{d}})^{T}}+(A+{A_{d}})X+BW+{W^{T}}{B^{T}}+{\bar {d}}{A_{d}}{A_{d}^{T}}$

Conclusion:

Given the resulting stabilizing control gain matrix $K=WX^{-1}$, it can be observed that the matrix is asymptotically stable from the time-delay system where this gain matrix was derived.

Implementation

• Example Code - A GitHub link that contains code (titled "DelayDependentTimeDelay.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

• Delay Independent Time-Delay Stabilization - Equivalent LMI for delay-independent time-delay stabilization.

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - A book co-authored by Guang-Ren Duan and Hai-Hua Yu.